High order Fuchsian equations for the square lattice Ising model:

On contact manifolds we describe a notion of (contact) finite-type for linear partial differential operators satisfying a natural condition on their leading terms. A large class of linear differential operators are of finite-type in this sense but are not well understood by currently available techniques. We resolve this in the following sense. For any such D we construct a partial connection r H on a (finite rank) vector bundle with the property that sections in the null space of D correspond bijectively, and via an explicit map, with sections parallel for the partial connection. It follows that the solution space of D is finite dimensional and bounded by the corank of the holonomy algebra of r H. The treatment is via a uniform procedure, even though in most cases no normal Cartan connection is available. The prolongations of a k th order linear differential operator between vector bundles arise by differentiating the given operator D : E ! F, and forming a new system comprising D along with auxiliary operators that capture some of this derived data. To exploit this effectively it is crucial to determine what part of this information should be retained, and then how best to manage it. With this understood, for many classes of operators the resulting prolonged operator can expose key properties of the original differential operator and its equation. Motivated by questions related to integrability and deformations of structure, a theory of overdetermined equations and prolonged systems was developed during the 1950s and 1960s by Goldschmidt, Spencer, and others (2, 17). Generally, results in these works are derived abstractly using jet bundle theory, and are severely restricted in the sense that they apply most readily to differential operators satisfying involutivity conditions. These features mean the theory can be difficult to apply. In the case that the given partial differential operator D : E ! F, has surjective symbol there is an effective algorithmic approach to this problem. The prolongations are constructed from the leading symbol �(D) : J k � 1 E ! F, where J k � 1 is the bundle of symmetric covariant tensors on M of rank k. At a point of M, denoting by K the kernel of �(D), the spaces K ` = ( J ` � 1 K) ( J k+` � 1 E), `� 0, capture spaces of new variables to be introduced, and the system closes up if K ` = 0 for

We give the Fuchsian linear differential equation satisfied by $\\chi^{(4)}$, the ``four-particle\'\' contribution to the susceptibility of the isotropic square lattice Ising model. This Fuchsian differential equation is deduced from a series expansion method introduced in two previous papers and is applied with some symmetries and tricks specific to $\\chi^{(4)}$. The corresponding order ten linear differential operator exhibits a large set of factorization properties. Among these factorizations one is highly remarkable: it corresponds to the fact that the two-particle contribution $\\chi^{(2)}$ is actually a solution of this order ten linear differential operator. This result, together with a similar one for the order seven differential operator corresponding to the three-particle contribution, $\\chi^{(3)}$, leads us to a conjecture on the structure of all the $ n$-particle contributions $ \\chi^{(n)}$.

This article is a sequel to the earlier articles, which describe the invertible ordinary differential operators and their generalizations. The generalizations are invertible mappings of filtered modules generated by one differentiation, and are called invertible D-operators. In particular, invertible ordinary linear differential operators, invertible linear difference operators with periodic coefficients, maps defined by unimodular matrices, and C-transformations of control systems are invertible D-operators. C-Transformations are those invertible transformations for which the variables of one system are expressed in terms of the variables of the other system and their derivatives.In the article we consider the invertible D-operators whose inverses are D-operators of the same type. In previous papers, a classification of invertible D-operators was obtained. Namely, a table of integers was associated to each invertible D-operator. These tables were described in a clear elementary-geometric language. Thus, to each invertible D-operator one assigns an elementary-geometric model, which is called a d-scheme of squares. The class of invertible D-operators having the same d-scheme was also described. In this paper, the invertible D-operators whose d-schemes consist of a single square are called unicellular. It is proved that any unicellular operator in some bases is given by an upper triangular matrix that differs from the identity matrix only by the first row. The main result is representation of the arbitrary invertible D-operator as a composition of unicellular operators. The minimum number of unicellular operators in such a composition is equal to the number of squares of the d-scheme of the original D-operator. As in previous papers, the used method is based on the description of d-schemes in the language of spectral sequences of algebraic complexes.The results obtained can be useful in the transformation and classification of control systems, in particular to describe flat systems.

Euler type partial differential operators on real analytic functions

An analogue of the irreducibility is developed for linear differential operators, or, more generally, noncommutative polynomials, and is applied to a few simple examples. Introduction. The question of the irreducibility of an ordinary linear differential operator is of interest in the Picard-Vessiot theory (see Kolchin [1, ?22]). Indeed, the operator is irreducible if and only if the Galois group is linearly irreducible. In this note we develop an Eisenstein criterion to help answer this question. Because a linear differential operator is a special type of noncommutative polynomial, it is more natural to phrase the result in this more general context. The result, and indeed the proof with only slight changes in notation, is valid for partial differential operators, or more generally, noncommutative polynomials in several variables. Because the hypothesis that the ring be principal is unnatural in this setting, and because the reader may make the necessary changes if he so desires, we confine our attention to the ordinary case. 1. The irreducibility criterion. By a differential ring M we shall mean a ring (associative, commutative, with unit) together with two commuting operators a and 3 such that a is an automorphism of R, 3 is additive, and 6(ab)=aa3b+6a.b for every a, b in S. 1[6] will denote the noncommutative ring of differential operators with coefficients in R, so that 1[6] is the set of 0 ai6i, with ai E $, n E N, and right multiplication is defined by 6 * a-=oa * + 3a. Evidently if or=id, then W is an ordinary differential ring with derivation operator 6. By a differential ideal of R we mean an ideal which is closed under the two operators a and 6. The smallest differential ideal which contains the set Ec$S is denoted by [2]. Received by the editors October 7, 1971. AMS 1970 subject classifications. Primary 12H05, 12E05; Secondary 13B25, 16A02.

The Hyers–Ulam stability constants of first order linear differential operators

Linear perturbations of differential of difference operators with polynomials as eigenfunctions

We use transfinite induction to show that if L is an epimorphism of a vector space V and maps a vector subspace W of V into a proper subspace of itself, then there is a smallest subspace E of V containing W such that L(E) E (or a minimal space of surjectivity or solvability) and we give examples where there are infinitely many distinct minimal spaces of solvability. We produce an example showing that if L 1 and L 2 are two epimorphisms of a vector space V which are endomorphisms of a proper subspace W of V such that LI(W) n L2(W is a proper subspace of W, then there may not exist a smallest subspace E of V containing W such that LI(E) E L2(E). While no nonconstant linear partial differential operator maps the field of meromorphic functions onto itself, we construct a locally convex topological vector space of formal power series containing the meromorphic functions such that every linear partial differential operator with constant coefficients maps this space linearly and continuously onto itself. Furthermore, we show that algebraically there is for every linear partial differential operator P(D) with constant coefficients a smallest extension E of the meromorphic functions in n complex variables, where

Tensor fields of arbitrary rank in multidimensional space are naturally extended to the concepts of divergence, gradient, curl, deformer operators, as well as their second-order superpositions. Two options for generalizing the rotor as an external product are presented. Differential operators of the second order that do not change the rank of the tensor to which they are applied are considered in detail. Square matrices are introduced, consisting of differential operators $${\text{Di}}{{{\text{v}}}_{{(l)}}}{\text{Gra}}{{{\text{d}}}_{{(k)}}}$$ , $${\text{Gra}}{{{\text{d}}}_{{(k)}}}{\text{Di}}{{{\text{v}}}_{{(l)}}}$$ , and their relationship is established. An explicit expression is written for the repeated operator rotor. All introduced generalized operators in particular cases agree in their properties with the corresponding classical operators in vector and tensor analysis.

We describe the spectrum of linear difference and differential operators with unbounded operator coefficients in weight spaces.

In this paper we construct local elementary solutions for linear differential operators P(x9 Dx) whose principal symbols are real and of simple characteristics and investigate their regularity properties using Sato's theory of the sheaf ^ (Sato DC^CAl). Throughout this paper we assume that the coefficients of differential operators are analytic. In §1 we prepare some theorems which extend the classical existence theorem of Cauchy-Kowalevsky in complex domain to cases of singular initial data. (Cf. Hamada [1].) In §2 we employ the results of §1 to construct local elementary solutions for Cauchy problems for (/-) hyperbolic operators. As an application of the method employed there, we construct a singular solution u(x) of P(x, Dx)u = Q whose singular support is very small. (See Theorem 2.8 for the precise meaning of this statement.) In §3 we construct a local elementary solution for a linear partial differential operator of real principal symbols with simple characteristics and decide where its singularities locate to conclude that the result of §2 on the existence of singular solutions with small singularities is the best possible one. Throughout this paper we denote by M an ra-dimensional real analytic manifold which we identify with a domain in R containing the origin.

Let P(z) be a polynomial of degree n with complex coefficients and consider the n–th order linear differential operator P(D). We show that the equation P(D)f = 0 has the Hyers–Ulam stability, if and only if the equation P(z) = 0 has no pure imaginary solution. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

This chapter contains results on generalized Hyers–Ulam stability, obtained by the authors, for linear differential equations, linear differential operators and partial differential equations in Banach spaces. As a consequence we improve some known estimates of the difference between the perturbed and the exact solution.

We consider nonlinear elliptic problems driven by a nonhomogeneous nonlinear differential operator. Using variational methods combined with Morse theory (critical groups), we prove two multiplicity results establishing three nontrivial smooth solutions. For the semilinear problem (linear differential operator), we produce four nontrivial smooth solutions. In the special case of the p-Laplacian differential operator, our framework of analysis incorporates equations which are resonant at infinity with respect to the principal eigenvalue.

Existence and uniqueness of advanced and retarded fundamental solutions (Green's functions) and of global solutions to the Cauchy problem is proved for a general class of first order linear differential operators on vector bundles over globally hyperbolic Lorentzian manifolds. This is a core ingredient to CAR-/CCR-algebraic constructions of quantum field theories on curved spacetimes, particularly for higher spin field equations.